Download Investigating Properties of Isosceles Trapezoids with the GSP: The

Investigating Properties of Isosceles Trapezoids with the GSP:
The case of a Pre-service Teacher
Adalira Sáenz-Ludlow Anna Athanasopoulou
[email protected]
[email protected]
Mathematics Department, University of North Carolina at Charlotte
Abstract
A four-month pilot teaching-experiment on the learning of geometry was conducted with two
pre-service teachers and two in-service teachers. The purpose of the pilot teaching-experiment
was to understand how learners, who already have some knowledge of geometric objects, are
able to reorganize this knowledge and broaden it in order to develop a better sense for the
geometric objects’ properties. The guiding principle for the tasks in this teaching-experiment was
to use semi-structured constructions using the Geometer’s Sketchpad (GSP) and open-ended
questions to foster learners’ freedom to explore, make conjectures, investigate them, and prove
them. The purpose of the task here analyzed was to investigate isosceles trapezoids and their
properties and the use of such properties to construct them. In this paper, we analyze the
interview in which a pre-service teacher (Susan) works on this task. Susan not only arrived at
several of the properties of isosceles trapezoids but also was able to prove them using her
knowledge of congruence of triangles.
Theoretical Rationale
Several studies in the teaching and learning of geometry (Choi-Koh, S. S., 1999; Mariotti, 2000;
Jiang, Z., 2002; Christou, Mousoulides, Pittalis, & Pitta-Pantazi 2004; and De Villiers, M., 2004,
among others) have used the GSP environment to mediate the interaction between learners and
geometric figures as well as the interaction between teacher and learners.
Jiang (2002), in his teaching-experiment, interviewed two pre-service teachers using the GSP
environment and he emphasizes that “after conjectures were made and tested, the subjects
continued GSP explorations to come up with insight for reasoning and proofs”. This teachingexperiment investigated the development of the geometric reasoning of two pre-service teachers
as well as the development of their ability to construct proofs. These teachers came to the
realization that students could also have a conceptual understanding of geometry if they were
given the opportunity to explore geometric figures and make and prove conjectures by
themselves rather than to commit to memory the proofs and solutions presented by the teacher.
Another teaching-experiment by Christou, Mousoulides, Pittalis, & Pitta-Pantazi (2004), using
prospective primary school teachers, concluded that the GSP was the bridge between exploration
and proof in geometry. In this teaching-experiment there was an emphasis in the phase before
proof where students constructed a kite using the properties of kites. Two of the three teachers
used the property of perpendicularity of its diagonals and the third used the property of two pairs
of adjacent equal sides. This knowledge was the starting point in the investigation of the kind of
quadrilateral formed by connecting the midpoints of adjacent sides and then proved that this
quadrilateral was a rectangle. This teaching-experiment shows that both the GSP and appropriate
guiding questions on the part of the teacher can propel students’ conceptualizations of
conjectures and their proofs.
De Villiers (2004) also using the GSP with prospective secondary-school mathematics teachers
asserts that they were able to start from different definitions of isosceles trapezoids and to prove
their properties and to construct different isosceles trapezoids according to the corresponding
definition. Teachers compared different constructions manifesting their preference for the use of
the “axis of symmetry” definition. Then they were also asked to define isosceles trapezoids and
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explore their properties. Using the GSP, students were able to construct generalizations as well
as specializations dragging vertices or line segments. In the conclusion of the teachingexperiment, the researcher indicates that the use of the GSP and structured activities (at the
corresponding Van Hiele level) provide valuable contexts for discoveries, explanations,
justifications, and proofs that could be incorporated in the high school geometry curriculum.
It appears as if the mediation of a dynamic software environment, like the GSP, allows students
to modify and refine their geometric knowledge while creating a channel for discussion and
communication between the teacher and students. Mariotti (2000) indicates that the role of the
teacher is fundamental to enhance mathematical discussion and to guide the evolution of
student’s construction of geometric meanings in a dynamic environment. She also points out that
according to her teaching-experiment; the classroom activities in a dynamic environment
allowed students to approach theoretical thinking in geometry. Choi-Koh (2001) also
investigated a secondary-school student’s development as he passed from the first to the last
level of the van Hiele’s model by means of the GSP environment and open-ended questions on
the part of the teacher. The student started at the intuitive level, and then continued to the
analytical level, and inductive level, to finally achieve the deductive level. The student used the
GSP as a tool not only for verification purposes but also for the construction of counterexamples.
According to this researcher, the geometry software provided an interactive environment
between the student and his constructions. Such an environment fostered the student’s
conjectures and the improvement of his geometric thinking as he tried to solve open-ended
problems prepared by the teacher. Choi-Koh also emphasizes the need of appropriate tasks for
the GSP environment both to link students’ intuitions and their ability to analyze geometric
objects to generate conjectures and to construct geometric proofs.
Methodology
Teaching experiment. The teaching-experiment methodology fostered long term interactions
between researchers and participant teachers to focus on their conceptual constructions and
cognitive manifestations. This pilot teaching-experiment lasted one academic semester and
consisted of one-to-one interviews with two pre-service teachers and two in-service teachers.
One researcher was the interviewer and the other was an active participant observer. The guiding
principle for the teaching-experiment was to use semi-structured tasks for the GSP environment
as well as open-ended questions to give learners the freedom to explore, to make conjectures, to
investigate them, and to prove them. Teachers were interviewed on a weekly basis and each
interview lasted 90 minutes. Here we concentrate on the analysis of one interview with Susan
(pseudonym), a pre-service secondary teacher.
A
C
Task. The aim of the task was to help Susan to investigate properties of
isosceles trapezoids through a semi-structured figure. Figure 1 was given to
her in the GSP environment. In this figure ABІІCD. The purpose of the task
Figure 1
was to analyze the figure by dragging points and segments and to anticipate
what kind of quadrilateral could be constructed from it and how it could be transformed into an
isosceles trapezoid. Another purpose of the task was to investigate and prove properties of
isosceles trapezoids and how the knowledge of these properties could be used to construct them
in the GSP environment.
O
D
B
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Analysis
Susan recognized the obvious triangles OAB and OCD and also a trapezoid if one were to
connect the segments AC and BD. She knew that a quadrilateral with one pair of parallel sides
was a trapezoid. The following question was about what types of trapezoids she knew, and she
answered “isosceles trapezoids”. She also took as well known facts that isosceles trapezoids have
the pair of non-parallel sides congruent and that the two pairs of base angles were also congruent
and she used these facts, as if they were definitions, to prove new found properties. The proofs
were based on the criteria of congruence of triangles and right triangles that we had reviewed at
the beginning of the teaching-experiment. Then, the interviewer probed Susan’s knowledge
about isosceles trapezoid and challenged her to construct an isosceles trapezoid using the given
Figure 1. Her very first reaction was to measure the non-parallel sides AC and BD but
immediately she switched into the property of equal diagonals and she measured them. Although
Susan described at once two properties of an isosceles trapezoid, she was not able to apply any of
them to construct an isosceles trapezoid using Figure 1. She tried to make the given trapezoid an
isosceles trapezoid dragging different vertices without complete success. Then Susan constructed
an isosceles trapezoid from scratch using the mirror process. She said that she used
S
P
this way of constructing an isosceles trapezoid in a project she did.
Susan constructed her own figure. She started from two line segments with a
T
common vertex, PR and RM (Figure 2). Then she constructed a perpendicular
line on RM at point M and using the GSP she mirrored these two segments over
Q
M
R
this perpendicular line. She completed the construction of the isosceles trapezoid
Figure 2
joining points P and S (see Figure 2). Then she constructed the diagonals and she measured
them. Dragging a vertex of the constructed isosceles trapezoid she confirmed her conjecture that
the diagonals were always equal (i.e., her figure was robust).
Not being able to construct the isosceles trapezoid using the properties she knew appears to be an
indication that her knowledge was only instrumental and not relational in Skemp’s (1987) sense.
She pushed herself to remember how she made this construction in her project. When she
remembered it, she made the construction using exactly the same process she did in the project.
It seemed that she did not know the property of isosceles trapezoids behind this construction (i.e.,
the bases of isosceles trapezoids have a common perpendicular bisector and this perpendicular
bisector could be used as a mirror or axis of symmetry). If she had known this property well, she
could have constructed the “mirror” she needed as the common perpendicular bisector to the
bases of the trapezoid into Figure 1 and transform that trapezoid into an isosceles one.
When Susan verified the congruence of the diagonals by dragging any vertex of the isosceles
trapezoid in the constructed figure the interviewer pushed her to jump into the following step and
prove the congruence of them. Susan expressed her thoughts as follows:
45 SUSAN:
Ok and I am going to color the triangles PTR and STQ just to color them. So these
two triangles triangle PTR and STQ are congruent [she gives no justification],
ok so that means again I know that the length of RT would be equal to the length of
TQ and again the diagonal RS can be RT plus TS and the diagonal TQ can be PT
plus TQ. RT is equal to TQ or the length of RT equal to the length of TQ, and the
length of ST is equal to PT. So that would make those two diagonals [RS and TQ] congruent.
(Figure 2)
The assertion that triangles PTR and STQ were congruent is not valid because the point T
(intersection of the diagonals) could or could not belong to the perpendicular bisector (i.e.,
mirror). Although the question was to prove that the diagonals were congruent, Susan
concentrated on the parts formed by the point of their intersection. To justify the congruence of
the triangles PTR and STQ she proposed to measure their areas. Susan’s persistence on
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measuring as a way of proving indicated a dependence on the software more than on geometric
reasoning. Therefore, the interviewer requested a proof independent of measurement. Susan
continued the analysis of the figure by considering point M as the midpoint of RQ (the midpoint
created by the mirror) and making TM the perpendicular bisector of RQ in order to conclude that
TQ was congruent to TR. When the interviewer asked her how she knew that point T (the
intersection of diagonals) was on the perpendicular bisector Susan was confused. For that reason
the interviewer, through dialogue and with appropriate questions, helped her to understand that
she had to compare the triangles PRQ and SQR since their sides PQ and SR were the diagonals
of the isosceles trapezoid. She accepted the hint.
88 SUSAN:
ok so I am looking at PRQ and SRQ… it’s cool. Oh! I get it, the light bulb just went on.
All right so ∆PTR and ∆STQ are congruent by side-angle-side (SAS), PR=SQ,
PRQ = SQR [she took as definition of isosceles trapezoids those trapezoids with
congruent base angles], and RQ=RQ. Thus, the sides [diagonals] PQ and SR are congruent.
Because Susan’s first idea was to compare the small triangles PTR and STQ, the interviewer
took the opportunity to ask her to prove the congruence of these triangles. Susan considered the
congruence of angles RPQ and QSR (as an implication of the congruence of ∆PRQ and ∆SRQ),
vertical angles PTR and STQ, and sides PR and SQ (because of the other part of her assumed
definition of isosceles trapezoids having simultaneously base angles congruent and non-parallel
sides congruent as inferred from her mirror-construction of the isosceles trapezoid). This
sequence did not fit in ASA and she did not know if it was correct. The instrumental
understanding came up one more time. It was clear that she was not able to make the connection
with the sum of the angles 180° in order to use the angle-side-angle criterion of congruence. So
the interviewer helped her to make this connection. Susan wrote the proof of the congruence of
the triangles and its implications TR=TQ and TP=TS concluding that the intersection of the
diagonals was on the perpendicular bisector of the bases in isosceles trapezoids. Prompted by the
interviewer, Susan summarized the properties of isosceles trapezoids.
Then the interviewer went further asking Susan to prove that if in a trapezoid the diagonals are
congruent, then the trapezoid is an isosceles trapezoid. It seems that such a proposition was
something strange in Susan’s mind because since the beginning she also took as definition of
S
isosceles trapezoids those trapezoids with congruent diagonals. She struggled P
trying to understand what she had to prove and finally she decided to construct a
T
new figure with two equal diagonals to form a quadrilateral. Then she realized
that such a quadrilateral was not a trapezoid and she constructed a parallel line
and verified that the figure do not have two parallel sides. At that point the
Q
interviewer clarified for her that the given was that the diagonals in a trapezoid R
Figure 3
were congruent and therefore she had to prove that under that condition the trapezoid would be
isosceles. Thus Susan abandoned the quadrilateral with the equal diagonals and returned to the
Figure 3.
Looking at Figure 3 she started a monologue as she was comparing the big
U
S
P
triangles PQR and SRQ and explained to herself that the triangles had only two V
sides congruent and there was no information about any angle. Susan could not
T
figure out how to prove it and she was repeating that “if the diagonals in a
trapezoid are equal then the trapezoid would be isosceles”. Then the interviewer
Q
guided her to figure out that she needed to construct right triangles to overcome R
Figure 4
the difficulty of the lack of angles. Finally, Susan constructed the appropriate
perpendicular lines forming a rectangle and so the right triangles RVS and QUP (Figure 4). After
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this construction, she proved that ∆RVS is congruent to ∆QUP using the congruence of the
diagonals RS=PQ and RV=UQ (perpendicular segments between parallel lines). Then she
proved that ∆RVP is congruent to ∆QUS (US=PV since VP+PS=PS+SU and, VR=UQ, and right
triangles). As an implication of this congruence she concluded that PR is congruent to SQ and
the trapezoid is an isosceles trapezoid.
Then the interviewer gave her a new proposition to prove: “if the intersection of the diagonals lie
on the perpendicular bisector of one of the bases of a trapezoid then the trapezoid is an
C
H
A
isosceles trapezoid”. Susan constructed the Figure 5 and proved that ∆GEB is
congruent to ∆GED (GE=GE, BE=DE just because it’s the midpoint, and then
G
BG=DG it’s the hypotenuse of Pythagorean Theorem). Then she proved that ∆AGH
is congruent to ∆CGH [ AGH = CGH because AGH= EGD vertical angles,
BGE= CGH vertical angles, and BGE = DGE as an implication of the B
E
congruent triangles GED and EGB, also HG=HG, and also the right angles GHA and
Figure 5
GHC are congruent]. Then, she implied that GA=GC. Finally, Susan explained that the point G
is equidistant from points A and C and therefore it lies on the perpendicular bisector of AC
making the trapezoid an isosceles trapezoid.
When Susan completed the last proof, the interviewer requested a different proof of the prior
proposition. Susan used the copy and paste facility and she got Figure 6. “Knowing that point G
lies on the perpendicular bisector of BD then triangle ∆GBD is an isosceles triangle A H 1 C
1
with congruent base angles. Since AC is parallel to BD it implies that the alternate
G
interior angles are congruent and therefore ∆GAC is isosceles too. Then the sum of
the parts of each diagonal are congruent, the diagonals are congruent, and the
1
trapezoid isosceles.” (Susan continues to use the property of the equality of the B 1
E
Fig ure 6
diagonals as a definition of isosceles trapezoid).
Susan went further and proved that ∆GBA and ∆GCD (AG=GC, GD=GB, and the vertical
angles AGB= CGD) are congruent implying that AB is congruent to CD and the trapezoid is
an isosceles trapezoid. Susan gave the last proofs easily. This fact that she was able to prove a
proposition and its converse appears to indicate that Susan have started to make the difference
between definition and properties of isosceles trapezoids.
Conclusion
The GSP provides opportunities to make dynamic constructions and it also fosters ways of
thinking and means of exploration that pencil-paper environments do not. It also mediates
learners’ dialogues with themselves and with others about the composition and decomposition of
geometric figures enhancing, at the same time, the visual modality of their thinking. The analysis
of the task presented in this paper indicates that Susan experienced the flexibility of the GSP
environment in ways that allowed her to deepen her knowledge about isosceles trapezoids.
The analysis also indicates that, step by step, the pre-service teacher investigated the properties
of isosceles trapezoids constructing different figures and exploring them in the GSP
environment. Then, she proved the properties using her previous knowledge about the
congruence of triangles and the relationships of the angles formed between parallel lines
intersected by a transversal. It also indicates that a good understanding of the properties of a
geometric figure furnishes the elements to make robust constructions in the GSP environment
(i.e., a robust construction is a construction of a figure which properties remain the same by
dragging any of its vertices or sides; that is to say that a robust construction generates a whole
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D
D
family of figures with the same characteristics; this is the meaning of the often used expression
“the figure passes the dragging test” when it is a robust figure).
In addition, the analysis also indicates that, at the beginning of the interview, Susan had only an
instrumental understanding (in Skemp’s sense) of some properties of the isosceles trapezoid
because she could not transform the trapezoid in Figure 1 into an isosceles one. Appropriate
questioning on the part of the interviewer helped the pre-service teacher to make connections
with her previous knowledge and to investigate several properties. Her improvement was also
manifested when, at the end of the interview, she gave different proofs of one of the properties.
Susan used six figures, the given and five that she constructed. The GSP also mediated Susan’s
explorations, her inferential thinking (inductive and deductive), her self-dialogues, and her
constructions as she acted and reacted upon them. It is worthwhile to notice that Susan had
different definitions of isosceles trapezoids because she considers their properties as definitions.
For example, one was that trapezoids are isosceles trapezoids if they have the non-parallel sides
and the pairs of base angles congruent; or those trapezoids which diagonals are equal. Properties
of the figure and their converses were unfamiliar to her.
During the four months of the teaching-experiment, the GSP became a useful tool to prepare
appropriate tasks and to pose open-ended questions according to the situation. Jones (2000) and
Arzarello et al. (2002) agree that the effectiveness of dynamic environment depends not only on
carefully designed tasks but also on the teacher’s sensitive guidance and emphasize the software
itself does not make an educational change happens but rather it is the teacher who has the
important role to guide students’ approach to theoretical thinking in the study of geometry.
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